A higher order p-adic class number formula

A HIGHER ORDER P -ADIC CLASS NUMBER FORMULA UNE FORMULE P -ADIQUE DU NOMBRE DES CLASSES D’ORDRE SUPERIEUR

arXiv:1208.0175v1 [math.NT] 1 Aug 2012

´ BLANCO-CHACON ´ IVAN

Abstract. We generalize a formula of Leopoldt which relates the p-adic regulator modulo p of a real abelian extension of Q with the value of the relative Dedekind zeta function at s = 2 − p. We use this generalization to give a statement on the non-vanishing modulo p of this relative zeta function at the point s = 1 under a mild condition.

´sum´ Re e. On g´en´eralise une formule de Leopoldt qui relie le r´egulateur p-adique modulo p d’une extension ab´elienne r´eelle de Q avec la valeur de la fonction zeta de Dedekind relative en s = 2 − p. On utilise cette g´en´eralisation pour donner un ´enonc´e sur la non-annulation modulo p de cette fonction zeta relative au point s = 1 sous une faible condition.

Let K/Q be a real abelian extension of degree g, class number h and discriminant d. Denote by ζK the Dedekind zeta function of K (ζ is the Riemann zeta function). In what follows, we will suppose that p > 3 is a prime number. For any integer number n ≥ 1, Bn,χ is the generalized Bernoulli number attached to n and χ. We fix once and for all an embedding alg of Qalg into Qalg denotes an algebraic closure of p , where for any field L, L L.

1. The p-adic class number formula mod p Let χ be a primitive character of Gal(K/Q) of conductor fχ and τ (χ) its Gauss sum. In [4], Leopoldt introduced the following p-adic analogue of the Partially supported by the project MTM2009-07024 of Spanish Ministerio de Ciencia e Innovaci´ on. 1

´ I. BLANCO-CHACON

2

complex value L(1, χ) fχ

τ (χ) X Lp (χ) := − χ(a) logp (1 − ξ a ) , fχ a=1 where ξ is a primitive fχ -root of 1 and logp is the Iwasawa logarithm. Proposition 1.1 (Leopoldt, [4], 1.8). Suppose that p ∤ fχ and that χ(−1) = 1. Then Bp−1,χ Lp (χ) ≡ χ(p) p (mod p2 ). p−1 The following definition is well known. Definition 1.2 (cf. Leopoldt, [4]). For any z ∈ Z∗p , the Fermat quotient of z (mod p), denoted by Qp (z), is the reminder of (z p−1 − 1)/p (mod p). The Fermat quotient satisfies the following property, whose proof is straightforward. Proposition 1.3. For any z ∈ Z∗p , logp (z) ≡ −pQp (z)

(mod p2 ).

Let {εk }g−1 k=1 be a fundamental system of units of OK , the ring of integers of K, and {σj }gj=1 a set of real embeddings of K into Qalg . Definition 1.4 (Leopoldt, [4]). The p-adic regulator mod p is R(p) (K) = det (Qp (σj (εk )))g−1 j,k=1 . As in the case of the standard p-adic regulator, which we denote by Rp (K), it is easy to check that R(p) (K) is well defined up to a sign. The following theorem is stated in [4] but it appears without a proof, which we give here. Theorem 1.5. Let K/Q be a real abelian extension and let p > 3 be a prime such that for any character χ of Gal (K/Q) of conductor fχ , p ∤ fχ .

UNE FORMULE P -ADIQUE DU NOMBRE DES CLASSES D’ORDRE SUPERIEUR 3

Then, with the above notations, there exists a fundamental system of units such that ζK (2 − p) 2g−1 hR(p) (K) √ ≡ (mod p). ζ(2 − p) d Proof. From the very definition of p-adic regulator mod p, for any fundamental system of units we have that Rp (K) ≡ (−p)g−1 R(p) (K) (mod pg ). From the p-adic class number formula, there exists a fundamental system of units such that 2g−1 hRp (K) Y √ = Lp (χ), d the product being taken over the set of non trivial characters of Gal(K/Q). Using Proposition 1.1, we obtain that Y Y (1.1) Lp (χ) ≡ (−p)g−1 L (2 − p; χ) (mod pg ). χ6=1

χ6=1

But the left hand side of (1.1) is congruent to

(−p)g−1 2g−1 hR(p) (K) √ (mod pg ). d 

Remark 1.6. In formula (3.8) of [4], the right hand side of the congruence appears with a potentially distinct sign. This is no contradiction, since regulators are defined up to a sign. 2. A p-adic class number formula mod pn and special values of p-adic relative zeta functions Given z ∈ Z∗p , write z = ω(z)hzi, where ω(z) is the unique (p − 1)-th root of 1 which is congruent to z modulo p. Denote by Qp,n (z) the truncation of the series −1 logp (1 + p˜ z ) (mod pn+1 ). p Definition 2.1. Let {εk }g−1 k=1 be a fundamental system of units of K and g {σj }j=1 a set of embeddings of K into Qalg . The p-adic regulator modulo pn is R(p,n) (K) = det (Qp,n (σj (εk )))1≤j,k≤g−1 .

´ I. BLANCO-CHACON

4

It is also easy to prove that R(p,n) (K) is also well defined up to a sign. The following lemma is obvious from the definition of Qp,n . Lemma 2.2. For any integer n ≥ 1, −pQp,n (z) ≡ logp (z) (mod pn+2 ). As a direct consequence of this lemma, we have the following result. Proposition 2.3. For any integer n ≥ 1, there exists a fundamental system of units such that Rp (K) ≡ (−p)g−1 R(p,n) (K) (mod pn+g ). ✷ Let Lp (s; χ) be the Kubota-Leopoldt p-adic L-function attached to χ (cf. [2], [1]). Proposition 2.4 (Kubota-Leopoldt, [2]). Let χ be a character of Gal(K/Q) of conductor fχ . Let p > 3 be a prime and suppose that p ∤ fχ . Then, for any s ∈ Zp , |Lp (1; χ) − Lp (1 − s; χ)|p ≤ |ps|p .

✷ Theorem 2.5 (Leopoldt, [3]). Lp (1; χ) = Lp (χ). ✷ Now, we state and prove our main result. Theorem 2.6. Let K/Q be a real abelian extension and p > 3 an unramified prime for K such that for any character χ of Gal (K/Q), p ∤ fχ . Then, there ∗ exists a fundamental system of units in OK such that for any n ≥ 1, ζK (1 − pn (p − 1)) 2g−1 hR(p,n) (K) √ ≡ ζ(1 − pn (p − 1)) d

(mod pn+1 ).

Proof. Given an integer n ≥ 1, by using Lemma 2.2 we have that Rp (K) ≡ (−p)g−1 R(p,n) (K) (mod pn+g ). By using Proposition 2.4 and Theorem 2.5,

UNE FORMULE P -ADIQUE DU NOMBRE DES CLASSES D’ORDRE SUPERIEUR 5

we obtain (2.1)

Y χ6=1

Lp (χ) ≡ pg−1

Y χ6=1

L (1 − pn (p − 1); χ)

(mod pn+g ).

By the p-adic class number formula, the left hand side of (2.1) equals (−p)g−1 2g−1 hR(p,n) (K) 2g−1 hRp (K) √ √ , which is congruent to (mod pn+g ). d d Dividing by pg−1 , the result follows.  With Proposition 2.6, we can formulate the following result on the nonvanishing mod p of the classical Dedekind zeta function Corollary 2.7. Let K/Q be a real abelian extension. Let p > 3 be an unramified prime such that (p, hk ) = 1 and Rp (K) ∈ pg−1 Z∗p . Then, for any n ≥ 1, ζK (1 − pn (p − 1)) p∤ . ζ(1 − pn (1 − p)) Proof. If Rp (K) ∈ pg−1 Z∗p , then, for any n ≥ 1, R(p,n) (K) ∈ Z∗p . Since p is unramified at K, (p, dK ) = 1. Since (p, hK ) = 1, the left hand side of the equality of Theorem 2.6 is a p-adic unit. Hence, the result follows.  Definition 2.8. (Iwasawa, [1]) Let K/Q be a real abelian extension of Q. Y The p-adic Dedekind zeta function of K is ζK,p(s) = Lp (1 − s; χ), where the product runs through the group of characters of Gal(K/Q). The function ζK,p is meromorphic with a simple pole at s = 0, which can be cancelled out dividing by Lp (1 − s; 1). The quotient is, hence, an analytic function denoted by ζK/Q,p. By passing to the limit in Corollary 2.7, we have the following result. Theorem 2.9. Let K/Q be a real abelian extension. Let p be an unramified prime such that (p, hk ) = 1 and Rp (K) ∈ pg−1 Z∗p . Then, ζK/Q,p(1) = 1. p

6

´ I. BLANCO-CHACON

Acknowledegements. I thank Pilar Bayer for carefully reading this note and for some valuable comments which allowed me to improve the earlier version, and Paloma Bengoechea, who has translated the abstract into French. References [1] K. Iwasawa: Lectures on p-adic L-functions. Annals of Mathematics Studies 74, 1972. [2] T. Kubota, H.W. Leopoldt: Eine p-adische Theorie der Zetawerte I. J. Reine und Angew. Math. 214/215 (1964), 328–339. [3] H.W. Leopoldt: Eine p-adische Theorie der Zetawerte II. J. Reine und Angew. Math. 274/275 (1975), 328–339. [4] H.W. Leopoldt: Zur Arithmetik in abelschen Zahlk¨ orpern. J. Reine und Angew. Math. 209 (1962), 54–71. ´ticas, Universidad de Barcelona, Gran Via de les Facultad de Matema Corts Catalanes, 585. 08007 Barcelona, Spain.